On posterior consistency in nonparametric regression problems

Choi, Tae-Yong; Schervish, Mark J.

doi:10.1016/j.jmva.2007.01.004

Cited by 102 publications

(136 citation statements)

References 14 publications

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“…For consistency in the Bayesian framework we utilize the theorem of Choi and Schervish (2007), and for asymptotic normality of the posterior we make use of Theorem 7.89 of Schervish (1995).…”

Section: Verification Of Posterior Normalitymentioning

confidence: 99%

On Bayesian asymptotics in stochastic differential equations with random effects

Maitra

Bhattacharya

2015

Statistics & Probability Letters

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Delattre et al. (2013) investigated asymptotic properties of the maximum likelihood estimator of the population parameters of the random effects associated with n independent stochastic differential equations (SDE's) assuming that the SDE's are independent and identical (iid).In this article, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution in the iid set-up as well as when the SDE's are independent but non-identical.

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Section: Verification Of Posterior Normalitymentioning

confidence: 99%

On Bayesian asymptotics in stochastic differential equations with random effects

Maitra

Bhattacharya

2015

Statistics & Probability Letters

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“…More recent developments have improved upon the classical results, avoiding the exceptional null set on which consistency may fail, see e.g. [10,6]. Other forms of posterior convergence have also been studied extensively, see e.g.…”

Section: Consistency Of the Bayes Estimatormentioning

confidence: 99%

Bayesian semiparametric Wiener system identification

2013

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We present a novel method for Wiener system identification. The method relies on a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process model for the static nonlinearity. We avoid making strong assumptions, such as monotonicity, on the nonlinear mapping. Stochastic disturbances, entering both as measurement noise and as process noise, are handled in a systematic manner. The nonparametric nature of the Gaussian process allows us to handle a wide range of nonlinearities without making problem-specific parameterizations. We also consider sparsity-promoting priors, based on generalized hyperbolic distributions, to automatically infer the order of the underlying dynamical system. We derive an inference algorithm based on an efficient particle Markov chain Monte Carlo method, referred to as particle Gibbs with ancestor sampling. The method is profiled on two challenging identification problems with good results. Blind Wiener system identification is handled as a special case.

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“…More recent developments have improved upon the classical results, avoiding the exceptional null set on which consistency may fail; see e.g. Choi and Schervish (2007); Barron et al (1999). Other forms of posterior convergence have also been studied extensively; see e.g.…”

Section: Consistency Of the Bayes Estimatormentioning

confidence: 99%

Particle filters and Markov chains for learning of dynamical systems

Lindsten

2013

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Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) methods provide computational tools for systematic inference and learning in complex dynamical systems, such as nonlinear and non-Gaussian state-space models. This thesis builds upon several methodological advances within these classes of Monte Carlo methods.Particular emphasis is placed on the combination of SMC and MCMC in so called particle MCMC algorithms. These algorithms rely on SMC for generating samples from the often highly autocorrelated state-trajectory. A specific particle MCMC algorithm, referred to as particle Gibbs with ancestor sampling (PGAS), is suggested. By making use of backward sampling ideas, albeit implemented in a forward-only fashion, PGAS enjoys good mixing even when using seemingly few particles in the underlying SMC sampler. This results in a computationally competitive particle MCMC algorithm. As illustrated in this thesis, PGAS is a useful tool for both Bayesian and frequentistic parameter inference as well as for state smoothing. The PGAS sampler is successfully applied to the classical problem of Wiener system identification, and it is also used for inference in the challenging class of non-Markovian latent variable models.Many nonlinear models encountered in practice contain some tractable substructure. As a second problem considered in this thesis, we develop Monte Carlo methods capable of exploiting such substructures to obtain more accurate estimators than what is provided otherwise. For the filtering problem, this can be done by using the well known RaoBlackwellized particle filter (RBPF). The RBPF is analysed in terms of asymptotic variance, resulting in an expression for the performance gain offered by Rao-Blackwellization. Furthermore, a Rao-Blackwellized particle smoother is derived, capable of addressing the smoothing problem in so called mixed linear/nonlinear state-space models. The idea of Rao-Blackwellization is also used to develop an online algorithm for Bayesian parameter inference in nonlinear state-space models with affine parameter dependencies.v Populärvetenskaplig sammanfattningMatematiska modeller av dynamiska förlopp används inom i stort sett alla tekniska och naturvetenskapliga discipliner. Till exempel, inom epidemiologi används modeller för att prediktera, dvs. förutsäga, spridningen av influensavirus inom en population. Antag att vi gör regelbundna observationer av hur många personer i populationen som är smittade. Baserat på denna information kan en modell användas för att prediktera antalet nya sjukdomsfall under, låt säga, nästkommande veckor. Den här typen av information möjliggör att en epidemi kan identifieras i ett tidigt skede, varpå åtgärder kan tas för att minska dess påverkan. Ett annat exempel är att prediktera hur hastigheten och orienteringen på ett flygplan påverkas då en viss styrsignal ställs ut på rodren, vilket är viktigt vid styrsystemdesign. Sådana prediktioner kräver en modell av flygplanets dynamik. Ytterligare ett exempel är att prediktera utvecklingen på e...

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On posterior consistency in nonparametric regression problems

Cited by 102 publications

References 14 publications

On Bayesian asymptotics in stochastic differential equations with random effects

On Bayesian asymptotics in stochastic differential equations with random effects

Bayesian semiparametric Wiener system identification

Particle filters and Markov chains for learning of dynamical systems

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